Symmetry-breaking bifurcations on cubic lattices

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Abstract
Steady-state symmetry-breaking bifurcations on the simple (SC), face-centred (FCC) and body-centred (BCC) cubic lattices are considered, corresponding to the 6-, 8- and 12-dimensional representations of the group . Methods of equivariant bifurcation theory are used to identify all primary solution branches and to determine their stability; branches with submaximal isotropy are generic for both the FCC and BCC lattices. Complete analysis of the local branching behaviour in the SC (three primary branches) and FCC (five primary branches) cases is given. The BCC case is substantially more complex, owing to the presence of a quadratic equivariant. A degeneracy that arises in the presence of an additional reflection symmetry is analysed first using a normal form truncated at third order. This problem, in which no quadratic equivariants are present, yields 10 primary branches with maximal isotropy and five with submaximal isotropy. The unfolding of the degeneracy is used to show that seven primary branches (six maximal and one submaximal) persist in the generic case, and to determine the form and properties of secondary branches. In certain cases higher-order terms are necessary. The study is motivated by the problem of pattern formation in three spatial dimensions, and extends earlier work by De Wit and co-workers.

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